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review derivation: curl curl identity

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
8 replace summation notation with vector notation
  1. 7575859310; locally 3948472:
    \(\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
  1. 7575859312; locally 2109231:
    \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
Nothing to split 7575859310:
7575859312:
7575859310:
7575859312:
2 replace curl with LeviCevita summation contravariant
  1. 7575859295; locally 1939485:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
  1. 7575859300; locally 9485482:
    \(\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
Nothing to split 7575859295:
7575859300:
7575859295:
7575859300:
5 substitute RHS of expr 1 into expr 2
  1. 7575859304; locally 2934842:
    \(\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}\)
    \(\)
  2. 7575859302; locally 2941319:
    \(\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
  1. 7575859306; locally 3949292:
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
failed 7575859304: failed
7575859302:
7575859306:
7575859304: N/A
7575859302:
7575859306:
7 simplify
  1. 7575859308; locally 3844221:
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
  1. 7575859310; locally 3948472:
    \(\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
Nothing to split 7575859308:
7575859310:
7575859308:
7575859310:
6 simplify
  1. 7575859306; locally 3949292:
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
  1. 7575859308; locally 3844221:
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
failed 7575859306:
7575859308:
7575859306:
7575859308:
3 replace curl with LeviCevita summation contravariant
  1. 7575859300; locally 9485482:
    \(\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
  1. 7575859302; locally 2941319:
    \(\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
Nothing to split 7575859300:
7575859302:
7575859300:
7575859302:
1 declare identity
  1. 7575859295; locally 1939485:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
no validation is available for declarations 7575859295:
7575859295:
9 claim LHS equals RHS
  1. 7575859312; locally 2109231:
    \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(\)
Nothing to split 7575859312:
7575859312:
4 declare identity
  1. 7575859304; locally 2934842:
    \(\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}\)
    \(\)
no validation is available for declarations 7575859304: failed
7575859304: N/A
Physics Derivation Graph: Steps for curl curl identity

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
4326 variable \vec{E}
\(\vec{E}\)
complex dimensionless electric field 9
1434 variable \hat{x}_n
\(\hat{x}_n\)
real dimensionless nth unit vector
  • str_note
1
1552 variable j
\(j\)
['integer']
index 7
7984 variable i
\(i\)
integer dimensionless index
  • str_note
1
8304 variable l
\(l\)
integer dimensionless index
  • str_note
3
6238 variable E
\(E\)
real dimensionless electric field 20
8349 variable \hat{x}_i
\(\hat{x}_i\)
real dimensionless ith unit vector
  • str_note
3
7930 variable m
\(m\)
integer dimensionless index
  • str_note
5
1567 variable i
\(i\)
['integer']
index 1
2380 variable \hat{x}_m
\(\hat{x}_m\)
real dimensionless mth unit vector
  • str_note
1
9690 variable k
\(k\)
integer dimensionless index
  • str_note
4
1592 variable n
\(n\)
integer
index 23
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